

Disclaimer, without doing a bunch of digging, my advice will be limited.
Some questions, are you using X11 or wayland?
Are you launching battle.net through a different method than the games that do work, i.e. are all your steam games running fine but you launch battle.net through lutris or bottles or something?
How long has your install been up and running?
It looks like the vulkan that works is a slightly higher version than the amd and rad ones (the first lines from the files you posted). Vulkan should be getting pulled in by Mesa, i would check to see if you have multiple versions of Mesa installed as well.
Depending on the cutoff year (when amdgpu as a default driver got replaced with RADV) , whether your cpu has a igpu (again, didn’t look up what gen those where), you could have 2 separate AMD drivers being installed, one for your igpu, and one for your dgpu.
Also the ones that are labeled (that don’t work but do output when you run those commands) could be used and pulled in because of X11, where the “unidentifed” is being used by wayland) i could be misreading, but i belive that command around pathing is more for X11 than wayland.
TLDR: the versions should be the first line in the texts you shared. Id figure out what is pulling and installing all 3 of those as deps for you to end up with them on your machine (unlabeled probably mesa, amd and rad probably drivers), and to research id look for some tools to debug and log what happens when you run a program so you can see the paths and LIBS hit.
Looks like you probably already hit this link: https://wiki.archlinux.org/title/Vulkan
Id check out this one too: https://wiki.archlinux.org/title/AMDGPU

Just to clarify, when mentioning problems with battle.net (in general), the issue you’re talking about is that when you click play for a game, afterwards the button shows “running” then switched back to “play” without anything popping up, correct?
If so, your issues probably aren’t related to anything on your machine except wine (as mentioned by other comments, don’t worry about drivers or vulkan or anything).
There is a known issue that when switching between wine versions, if that exact scenario occurs, you have to blow away the wine prefix.
For some games this is harmless, some save config files or saves for players in the prefix.
Instead of trying to lock drivers or mesa or vulkan to specific versions, instead lock whatever version of wine you have set for battle.net in lutrus to a static version. That includes not using proton-experiemental or proton-ge latest, but instead a set version like proton-ge10.26 (or whatever).